### Quantum Gravity Phenomenology

General Relativity and Quantum Mechanics are both extremely successful in their respective domains of applicability,
but these are very different: loosely speaking, the former applies on very large scales, from the Solar System to
cosmological distances; the latter becomes relevant in the microscopic world. There are also deep differences in the
structure of the two theories: one is classical, and posits that the geometry of spacetime is dynamical, while the
other is a probabilistic theory and relies on fixed background structures for the very interpretation of its
predictions. Therefore it should not surprise us that attempts to describe one theory using the language of the other
encounter formidable difficulties. Despite decades of attempts, we still have no quantum theory of cosmology; the basic
problems with the interpretation of a `wavefunction of the universe' remain as obscure as ever, despite recognition long
ago of the profound difficulty they present. Analogously, all attempts to interpret General Relativity as the classical
limit of a quantum field theory and to make it into a quantum theory of gravitons have failed, typically frustrated by
the non-renormalizability of GR, which makes it an ill-defined quantum theory. Considering the vastly different regimes
of applicability of the two theories, these problems might seem purely academic, but, while not easy to access
experimentally, GR and Quantum Mechanics do have an overlapping regime where they should both be taken into account. In
particular, this regime is relevant for an understanding of the first instants of the evolution of the Universe.
Several approaches to the creation of quantum gravity have been investigated, but so far none has produced a fully
satisfactory outcome and none has found support in experimental data. All the most studied scenarios for quantum gravity
do predict new phenomena, but the most natural candidate for the characteristic scale of these new phenomena is the
ultrashort Planck length

*L*_{p} ~ 10^{-35} m, and this makes the magnitude of the new effects far
too small for most experimental testing. For example, Loop Quantum Gravity predicts a cosmological bounce when the
universe approaches a big crunch, but this happens when the size of the universe is comparable to the Planck length.
Exceptions to this problem of untestability have started to emerge over the last 15 years. For example, one of the most
natural implications of quantum gravity is the replacement of spacetime, which is a pseudo-Riemannian manifold in GR,
with some kind of quantum

*i.e.* noncommutative structure. In some models, this affects the propagation of
particles in vacuum at levels that, while minute in absolute terms, are within the reach of certain very sensitive
experiments. These effects are now being tested with astrophysical observations, for example of gamma ray bursts, which
emit particles over a large range of energies. Since these sources are at cosmological distances, any difference in the
way particles with different energy or polarization propagate will accumulate and build up to possibly observable levels
[

1]. Similarly effects of this kind can be tested in table-top experiments, like atomic interferometers, as I
showed in [

2].
Another example of testable effects regards the possibility that spacetime (or space) undergoes a `dimensional
reduction' at ultrashort distances, becoming a lower- or higher-dimensional (or even a fractal) structure. This feature
is present in a variety of different approaches to quantum gravity, from Asymptotic Safety to Causal Dynamical
Triangulations, and most of them point towards a value of 2 for the effective dimensionality at small scales. This
`dimensional reduction' might be invoked as an explanation for the scale-invariant spectrum of scalar fluctuations
observed in the CMB, and makes predictions regarding the spectrum of tensor fluctuations [

3].
Recently, there has been a significant leap forward (to which I contributed [

4]) in the formalization of
effective models capturing the `first order' corrections to the propagation of particles that we might expect to arise
from quantum gravity. This goes under the name of `Relative Locality' [

5]. It is based on the
idea that in a quantum spacetime the notion of localization has to be relaxed and to become a relative notion, much like
simultaneity in special relativity. This simple idea produces a host of possibly testable new physical effects of which
we have barely started to scratch the surface.
At the moment I am interested in the connections between Relative Locality and the mathematics of Hopf Algebras (also
called Quantum Groups). I contributed to the first clarification of this in my paper [

6]. A possible
further exploration of these connections would be to study the

*SU*_{q}(2) Quantum Group from the perspective of Relative
Locality. This group has an interesting role in spinfoam models, where deforming the structure group

*SU(2)* into

*SU*_{q}(2) leads to inclusion of a positive cosmological constant in the theory [

6]. It has been argued
that this group implies a fundamental limitation on the accuracy of angular measurements. It looks natural to explore
this conjecture in the context of Relative Locality.